Re: Let's look at what's there...
> 1. There are 4 pistons, which can only engage with the 4 landing legs on the Falcon 9 core.
I suspect they engage similar to the hold-down clamps used during a launch, holding the body, not the legs.
> 3. [...] but it's wheels (or treads) aren't visible. [...]
To the left and right of center there appears to be treads.
> Add it all up, and it seems the robot's purpose is to move freshly-landed cores.
My guess is that it's to stabilize the landed booster stage during transport, possibly using electromagnets to secure the robot to the deck.